Raising bilingual children as non-native speakers is possible and can be very rewarding. This is my story of how I raised my child to be bilingual (English and Hebrew) as a non-native English speaker myself.
My son is three and a half years old today and can understand both English and Hebrew equally as well, though he refuses to speak in English :-) (more on that later). As for me, although I am not a native English speaker, my grammar, vocabulary, and pronunciation are very good.
First, a small disclaimer - unless mentioned otherwise, the views expressed in this article are my personal views stemming from my experience and should not be taken as scientific advice.
Why raise a bilingual child?
Being bilingual has been shown by scientists to have cognitive , educational, and health  benefits. Yet, to be honest, those benefits are a bonus as far as I'm concerned. My motivation to raise my child as a bilingual speaker had more to do with the social and economic benefits such as:
Knowing more than one language allows one to connect with a wide range of people on a deep level, thus boosting one's confidence in social interactions drastically.
Knowing a second language (and specifically English) makes you more competitive in the job market.
After all, as parents, we want to give our children the opportunities and tools we never had.
How to raise a bilingual child as a non-native speaker
Before my son was born I researched the subject myself - from reading articles and blog posts online to talking to other people (non-native speakers) who were raising their children as bilingual. I came up with a few rules that I followed religiously:
I prefer "simple" and correct sentences over "complex" sentences with mistakes. This is probably the most challenging part of all, and it's one of those easier said than done things. First, a bit about the rationale. If you make grammar and pronunciation mistakes in front of your child repeatedly then you'll be teaching those mistakes to your child (assuming you are the only person speaking with him in your non-native language), and it will be hard for your child to unlearn those mistakes in the future. On the other hand, your child will have no problem enriching his/her second language in the future when it's based on a good foundation of grammar and pronunciation.
I strive to always use my non-native language when speaking with my child. I say "strive" because we are all humans after all. Don't feel bad if you switch to your native language, just be sure to switch back when you notice.
My wife uses our native language when speaking with my son. While it's important for me that my son will learn a second language, I did want to make sure he mastered his native language at home. I didn't want him to get into the local kindergarten and have problems communicating with the other children.
I use my native language to speak with everyone else (including my wife, and my son's grandparents). This was done for a couple of reasons: a) as mentioned above, it was important for me to make sure he mastered his native language at home; b) other people will not necessarily follow rule #1 above.
Language learning is a never-ending, beautiful and rewarding process. Over the past three and one-half years not only did my son learn two languages, but I actually improved my fluency and vocabulary along the way too. Every day I am forced to look up words I do not know, for the sole purpose of being able to speak with my son. It's FUN! It's like we have our own secret language.
As for my son, it is normal that he refuses to speak in English (except for words that he only knew in English). This happens with a lot of bilingual children so I never insisted on changing that. I believe it is important for his future success that I keep this process joyful for him.
At the end of the day, even though my son is not yet four years old, he understands English and Hebrew equally. The part that I get to play in his progress gives me great satisfaction. If you believe you can follow the rules above, I strongly recommend you do so!
Elad Amrani, Co-Founder of Lingos.
 Marian V, Shook A. The cognitive benefits of being bilingual. Cerebrum. 2012 Sep; 2012:13. Epub 2012 Oct 31. PMID: 23447799; PMCID: PMC3583091. The Cognitive Benefits of Being Bilingual
 Manfred Spitzer, Bilingual benefits in education and health, Trends in Neuroscience and Education, Volume 5, Issue 2, 2016, Pages 67-76, ISSN 2211-9493, https://doi.org/10.1016/j.tine.2016.07.004. https://www.sciencedirect.com/science/article/pii/S2211949316300163) Bilingual benefits in education and health